I just had a baby in Taiwan roughly a year ago, so I will tell you how that went for my wife. I am American and my wife is Taiwanese.
I would say Taiwan would be best as the doctors can explain everything to her in Chinese and most doctors can also explain it to you in English. Cost is another huge thing. Here the monthly pregnancy checkups just cost a couple hundred. The doctor we went to did an ultrasound every visit. I heard at some hospitals they don’t do that. Some of the tests you might have to pay out of pocket for. The most expensive test cost 30k, I think it was a down syndrome test. There were cheaper options, but my wife wanted that one. During delivery you can have your doctor come in and deliver the baby. Not sure if it costs extra. I actually feel bad for our doctor because it seemed like she was always at the hospital.
Delivery costs are another thing. We went to a regular hospital and did everything. I know there are birthing centers, and I believe they cost more. We paid close to 120k. That was for a c-section, pain medicine, a 2 person room, and some special bandages and medicine for a c-section that NHI doesn’t cover. The guest chair/bed in a 2 person room is very uncomfortable. The single room chair/bed looks a lot more comfy.
You will also be expected to do a lot of things for her when she is staying in the hospital. For example if she needs a pad changed or the catheter pee bag emptied, the nurses will show you the first time and then you will be expected to do it the rest of the time. If you ask though they will help.
Check with your wife to see if she has private insurance as well. My wife did and was able to get reimbursed for everything.
As you’re trying to get pregnant, you need to decide if you are going to one of those afterbirth hotel places. (I can’t remember their proper name) The popular ones book up fast. If you do decide you are going to one make sure you book early. We ended up not going to one and instead just had some healthy food delivered to the house. Which was very convenient as I didn’t have to worry about cooking for her.
Getting your kid a US citizenship is very easy. First get an English birth certificate from the hospital with both your wife’s and your name on it and can a few copies while your at it. Then make an appointment at the AIT. They book up pretty fast so you might need to book a month or two in advance. Gather up all the necessary paperwork to prove you are American. There’s a checklist on the AIT’s website. Then you, your wife, and kid go to the AIT. The whole process takes about 2 hours and you will have your kids passport and birth aboard report in about 2 months. After you get the kids passport make another AIT appointment to get a copy of your passport and his notarized for his SSN. The AIT will likely provide this free to you. (they did for me at least). Fill out the SSN application and mail the documents to the SSN office in the Philippines. You have a choice on where to receive the SSN card. In Taiwan which will take a few months, or send it to someone you know in America, like your parents, that takes about 2 weeks.
I’m sure there are lots of things I forgot to mention, so feel free to ask any questions and I will do my best to answer.